Continuing from Biggs’s Algebraic Graph Theory, we discuss the properties of the Laplacian Matrix of a graph and how it relates to the tree number.
The cornerstone in local theory of Banach spaces is Dvoretzky’s theorem, which asserts that almost euclidean structure is locally present in any high-dimensional normed space. The random version of this remarkable phenomenon was put forth by V. Milman in 70’s, who employed the concentration of measure on the sphere. Purpose of the talk is to present how Gaussian tools from high-dimensional probability (e.g., Gaussian convexity, hypercontractivity, superconcentration) can be exploited for obtaining optimal results in random forms of Dvoretzky’s theorem. Based on joint work(s) with Grigoris Paouris and Konstantin Tikhomirov.
Cosmetic surgeries (purely cosmetic surgeries) are two distinct surgeries on a knot that produce homeomorphic 3-manifolds (as oriented manifolds). It is one of the ways Dehn surgeries on knots could fail to be unique. Gordon conjectured that there are no nontrivial purely cosmetic surgeries on nontrivial knots in S^3. We will recap the progress of the problem over time, and mainly discuss Ni and Wu's results in their paper "Cosmetic surgeries on knots in S^3".
Two classic questions -- the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem -- both focus on the distance, which is a simple two point configuration. When studying the Falconer distance problem, a geometric averaging operator, namely the spherical averaging operator, arises naturally. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will give a brief introduction to the motivating point configuration questions and then report on some novel geometric averaging operators and their mapping properties.
The malaria parasite Plasmodium falciparum requires a vertebrate host, such as a human, and a vector host, the Anopheles mosquito, to complete a full life cycle. The portion of the life cycle in the mosquito harbors both the only time of sexual reproduction, expanding genetic complexity, and the most severe bottlenecks experienced, restricting genetic diversity, across the entire parasite life cycle. In previous work, we developed a two-stage stochastic model of parasite diversity within a mosquito, and demonstrated the importance of heterogeneity amongst parasite dynamics across a population of mosquitoes. Here, we focus on the parasite dynamics component to evaluate the first appearance of sporozoites, which is key for determining the time at which mosquitoes first become infectious. We use Bayesian inference techniques with simple models of within-mosquito parasite dynamics coupled with experimental data to estimate a posterior distribution of parameters. We determine that growth rate and the bursting function are key to the timing of first infectiousness, a key epidemiological parameter.
We consider the problem of finding on a given bounded and smooth
Euclidean domain \Omega of dimension n ≥ 3 a complete conformally flat metric whose Schouten
curvature A satisfies some equation of the form f(\lambda(-A)) =1. This generalizes a problem
considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of
locally Lipschitz solutions. We also show that the Lipschitz regularity is in general optimal.
The dual of the cone of non-negative quadratics (on a variety) is included in the dual of the cone of sums of squares. Moreover, all (points which span) extreme rays of the dual cone of non-negative quadratics is point evaluations on real points of the variety. Therefore, we are interested in extreme rays of the dual cone of sums of squares which do not come from point evaluations. The dual cone of sums of squares on a variety is called the Hankel spectrahetron and the smallest rank of extreme rays which do not come from point evaluations is called Hankel index of the variety. In this talk, I will introduce some algebraic (or geometric) properties which control the Hankel index of varieties and compute the Hankel index of rational curves obtained by projecting rational normal curve away from a point (which has almost minimal degree).
A classical theorem of Lie and Tresse states that the algebra of differential invariants of a Lie group or (suitable) Lie pseudo-group action is finitely generated. I will present a fully constructive algorithm, based on the equivariant method of moving frames, that reveals the full structure of such non-commutative differential algebras, and, in particular, pinpoints generating sets of differential invariants as well as their differential syzygies. Some applications and outstanding issues will be discussed.
The past few years have seen the advent of big data, which brings unprecedented convenience to our daily life. Meanwhile, from a computational point of view, a central question arises amid the exploding amount of data: how to tame big data in an economic and efficient way. In the context of matrix computations, the question consists in the ability to handle large dense matrices. In this talk, I will first introduce data-sparse hierarchical representations for dense matrices. Then I will present recent development of a new data-driven algorithm called SMASH to operate dense matrices efficiently in the most general setting. The new method not only outperforms existing algorithms but also works in high dimensions. Various experiments will be provided to justify the advantages of the new method.
